Write the differential equation for angular shm?
Question: Write the differential equation for angular shm?
The differential equation for angular Simple Harmonic Motion (SHM) can be written as:
d²θ/dt² + (k/m)θ = 0
where θ is the angular displacement from the mean position, t is time, k is the angular spring constant, and m is the moment of inertia of the system.
This equation is similar to the equation for linear SHM, which is d²x/dt² + (k/m)x = 0, where x is the displacement from the mean position. However, in the case of angular SHM, the displacement is measured in radians rather than meters.
The solution to this differential equation is a sinusoidal function of time, with an amplitude, frequency, and phase determined by the initial conditions of the system. The period of the motion is given by T = 2π√(m/k), where m is the moment of inertia and k is the angular spring constant. The frequency of the motion is f = 1/T, and the angular velocity is ω = 2πf.
Angular SHM is a common phenomenon in many physical systems, including pendulums, torsion pendulums, and molecular vibrations. It is an important concept in the study of physics and engineering, and is used in a wide range of applications, including the design of mechanical systems, the analysis of molecular dynamics, and the study of astronomical phenomena.
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